I can’t make sense of the part with R-world and L-world. You assign probabilities to your possible actions (by what rule?) then do arithmetic on them to decide which action to take (why does that depend on probabilities of actions?) then rotate the picture and find that actions are correlated with hidden facts (how can such correlation happen?) It looks like this metaphor doesn’t work very well for decision-making, or we’re using it wrong.
Well… I agree with all of the “that’s peculiar” implications there. To answer your question:
The assignment of probabilities to actions doesn’t influence the final decision here. We just need to assign probabilities to everything. They could be anything, and the decision would come out the same.
The magic correlation is definitely weird. Before I worked out an example for this post, I thought I had a rough idea of what Jeffrey-Bolker rotation does to the probabilities and utilities, but I was wrong.
I see the epistemic status of this as “counterintuitive fact” rather than “using the metaphor wrong”. The vector-valued measure is just a way to visualize it. You can set up axioms in which the Jeffrey-Bolker rotation is impossible (like the Savage axioms), but in my opinion they’re cheating to rule it out. In any case, this weirdness clearly follows from the Jeffrey-Bolker axioms of decision theory.
The assignment of probabilities to actions doesn’t influence the final decision here. We just need to assign probabilities to everything. They could be anything, and the decision would come out the same.
Aren’t there meaningful constraints here? If I think it’s equally likely that I’m in L-world and R-world and that this is independent of my action, then I have the constraint that P(Left, L-world)=P(Left, R-world) and another constraint that P(Right, L-world)=P(Right, R-world), and if I haven’t decided yet then I have a constraint that P>0 (since at my present state of knowledge I could take any of the actions). But beyond that, positive linear scalings are irrelevant.
I can’t make sense of the part with R-world and L-world. You assign probabilities to your possible actions (by what rule?) then do arithmetic on them to decide which action to take (why does that depend on probabilities of actions?) then rotate the picture and find that actions are correlated with hidden facts (how can such correlation happen?) It looks like this metaphor doesn’t work very well for decision-making, or we’re using it wrong.
Well… I agree with all of the “that’s peculiar” implications there. To answer your question:
The assignment of probabilities to actions doesn’t influence the final decision here. We just need to assign probabilities to everything. They could be anything, and the decision would come out the same.
The magic correlation is definitely weird. Before I worked out an example for this post, I thought I had a rough idea of what Jeffrey-Bolker rotation does to the probabilities and utilities, but I was wrong.
I see the epistemic status of this as “counterintuitive fact” rather than “using the metaphor wrong”. The vector-valued measure is just a way to visualize it. You can set up axioms in which the Jeffrey-Bolker rotation is impossible (like the Savage axioms), but in my opinion they’re cheating to rule it out. In any case, this weirdness clearly follows from the Jeffrey-Bolker axioms of decision theory.
Aren’t there meaningful constraints here? If I think it’s equally likely that I’m in L-world and R-world and that this is independent of my action, then I have the constraint that P(Left, L-world)=P(Left, R-world) and another constraint that P(Right, L-world)=P(Right, R-world), and if I haven’t decided yet then I have a constraint that P>0 (since at my present state of knowledge I could take any of the actions). But beyond that, positive linear scalings are irrelevant.